Question

Suppose the stress tensor is
T =
x yz 2
yz y x
2 x z
(a) Assuming there are no body forces, explain why it is not possible that the
material is at rest.
(b) What would the body force need to be so the material is at rest?

Answer 1

a) The total force applied on a body can be given as,

where

Even if the body forces are zero, there will be surface forces. So, the total forces will not be zero. That means, the body possesses acceleration. The accelerating body will be under motion.

Hence, zero body forces does not mean that the body is at rest.

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b)

From the equation for total force on a body,

For the body to be at rest, the total force acting on it should be zero. That means,

Applying this condition in the equation we get,

or,

It means that, for the body to be at rest, the body forces should have an equal magnitude of surface forces and the body forces should have a direction opposite to that of the surface forces.